Why are my UV islands splitting apart when I move them?
It looks like this is a coincidence of a few things. When trying this on my machine, I can replicate the behavior if I do the following things at the same time:


I am in UV Sync Selection Mode
I have select UV islands enabled
I try to move the UVs not by using g, but by click-dragging with the mouse instead


check the five icons on the bottom right of this screenshot:



the one on the very right is the UV islands selection mode, it is enabled. You only see this if the fifth icon from the right, the one that looks like an arrow over a cube, is disabled. This icon toggles the UV sync option. If it is off like now, you only see UVs for vertices which are selected in the 3D viewport. So make sure that in 3D View, all vertices are selected.

Now the crucial part:

The click-dragging actually updates your selection before the moving starts in the UV editor, and selects (in this case) only the one island under the mouse cursor. This behavior is sometimes called 'tweak mode'. If you select first, and then hit g to go to translation mode, the whole stuff translates, no matter if it is connected or not.